Multiparticle one-loop amplitudes and S-duality in closed superstring theory

TL;DR

This work computes explicit one-loop five-particle amplitudes in Type II superstring theory using the pure spinor formalism and reveals that the Type IIB result can be written as a bilinear in Yang–Mills tree amplitudes, while Type IIA contains additional parity-allowed terms. By performing a careful low-energy expansion in powers of $\alpha'$ up to $D^{2k}R^5$ for $0\le k\le 5$, the authors uncover a rich pattern of kinematic invariants and lattice sums that feed into SL$(2,\mathbb{Z})$-invariant modular functions, constraining moduli dependence via S-duality. They analyze both $U(1)$-conserving and $U(1)$-violating amplitudes, establishing how the perturbative coefficients align with duality expectations and identifying new one-loop structures (including primed matrices like $M'_{7}$) that have no tree-level counterparts. The six-point analytic sector is partially constrained, indicating vanishing $R^6$ and $D^4R^6$ terms in ten dimensions and suggesting nontrivial lower-dimensional remnants and a broader modular framework governing higher-point interactions. Overall, the results illuminate how pure spinor amplitudes, SYM tree data, and world-sheet integrals organize into duality-consistent effective actions, guiding future explorations of higher-point and higher-genus string amplitudes and their nonperturbative moduli dependence.

Abstract

Explicit expressions for one-loop five supergraviton scattering amplitudes in both type II superstring theories are determined by making use of the pure spinor formalism. The type IIB amplitude can be expressed in terms of a doubling of ten-dimensional super Yang--Mills tree amplitude, while the type IIA amplitude has additional pieces that cannot be expressed in that manner. We evaluate the coefficients of terms in the analytic part of the low energy expansion of the amplitude, which correspond to a series of terms in an effective action of the schematic form D^{2k}R^5 for 0\le k \le 5 (where R is the Riemann curvature). Comparison with earlier analyses of the tree amplitudes and of the four-particle one-loop amplitude leads to an interesting extension of the action of SL(2,Z) S-duality on the moduli-dependent coefficients in the type IIB theory. We also investigate closed-string five-particle amplitudes that violate conservation of the U(1) R-symmetry charge -- processes that are forbidden in supergravity. The coefficients of their low energy expansion are shown to agree with S-duality systematics. A less detailed analysis is also given of the six-point function, resulting in the vanishing of the analytic parts of the R^6 and D^4 R^6 interactions in the ten-dimensional effective action, but not in lower dimensions.

Figures (4)

• Figure 1: Three different topologies of five-point integrals: Directed dashed lines represent both holomorphic and antiholomorphic derivatives $\partial \ln \chi_{ij}$ and $\tilde{\partial} \ln \chi_{ij}$.
• Figure 2: Momentum conservation at vertices of world-sheet diagrams following from $z_i$ integration.
• Figure 3: Graphical formulation of the Laplace equation: The first diagram on the right-hand side originates from the $\tau_2^{-1}$ factor in $\partial \bar{\partial} \ln \chi_{ij}\sim (\delta^2(z_{ij}) - \tau_2^{-1})$, the second diagram is due to the delta function admixture.
• Figure 4: Possible topologies of six-particle integrals with four $\ln \chi$ derivatives where both the $\partial \ln \chi_{ij}$'s and the $\bar{\partial} \ln \chi_{ij}$'s carry four different labels. The integration measure for the above expressions can be found in the first line of (\ref{['6ptcorr']}). A distinction between $\partial$ and $\tilde{\partial}$ is not needed since dashed lines with alike derivatives never end on the same vertex